GROWTH CURVE AND BODY WEIGHT IN GOETTINGEN MINIPIGS A PHENOTYPIC AND GENETIC STUDY
Dissertation for the Doctoral Degree
at the Faculty of Agricultural Sciences, Georg-August-University Göttingen
presented by Friederike Köhn
born in Berlin
Göttingen, September 2007
D 7
1. Referee: Henner Simianer 2. Co-referee: Peter Bollen
Date of disputation: 15. November 2007
“I am very proud to be called a pig.
It stands for pride, integrity and guts.”
Ronald Reagan (1911-2004)
I would like to thank
Henner Simianer for working as main supervisor. Your great support and your never ending ideas were a big help and motivation.
Peter Bollen for acting as second supervisor.
Ellegaard Göttingen Minipigs ApS in Dalmose, Denmark, for financial support and for providing the minipig weight data. Thank you for a very interesting and motivating week in October 2005. Special thanks to Jens and Lars Ellegaard with families for the support and to Nanna Grand who always answered all my questions, no matter how stupid they were.
Špela Malovrh from the Biotechnical Faculty in Domžale, University of Ljubljana, Slovenia, for her essential help with random regression models and VCE. Without you I would have been completely discouraged.
Reza and Helge for always having time to help me with all problems concerning statistics and genetics. Thank you Reza for all your entertaining Persian stories.
Christine Flury for her good guidance in working with the minipig database and for helpful advices in all situations.
Oskar Lippstreu for the help with weighing the minipigs in Relliehausen and for answering all management questions.
The keepers Jochen, Dieter, Jürgen, Uwe, Sascha, Carina and Rebekka for taking care about my minis and relaxing talks.
Tamina, Janet and Rebecca for motivation, relaxation and interesting talks. You made every boring working day much more colorful.
All colleagues from the Institute for help, funny lunches and inspiration.
Emma, Elvis and Spotty for a perfect daily relaxation and for animating the lifeless data.
Jan for his patience and motivation. You are enriching my life.
My sister Steffi for a perfect childhood and for surviving all my bad moods.
Meinen Eltern danke ich für ihre unglaubliche Unterstützung und ihre bedingungslose Liebe. Ihr habt mir immer den besten Weg gezeigt und mich nie allein gelassen, wenn ich trotzdem einen anderen gewählt habe. Ich danke Euch für eine fantastische Kindheit, eine aufregende Jugend und ein entspanntes Erwachsenwerden.
Table of Contents
Zusammenfassung 7
Summary 9
1st CHAPTER General introduction 11
2nd CHAPTER Modeling the growth of Goettingen minipig 17
3rd CHAPTER Estimation of genetic parameters for body weight of the Goettingen minipig with random regression models
36
4th CHAPTER Breeding for low body weight in Goettingen minipigs 51
5th CHAPTER General discussion 71
Zusammenfassung
Die vorliegende Arbeit befasst sich mit der Untersuchung einer möglichen Gewichtsredu- zierung beim Göttinger Minischwein und der Ermittlung einer bestmöglichen Selektions- strategie für dieses Merkmal. Die Ziele waren die Analyse der Körpergewichtsentwicklung des Göttinger Minischweins und die Schätzung genetischer Parameter für Körpergewicht.
Weiterhin wurden verschiedene Selektionsstrategien untersucht, um anschließend ein neues Zuchtschema aufstellen zu können.
In einem ersten Schritt wurde das Wachstum der Minischweine untersucht. Die Gewichts- entwicklung der Minischweine wurde durch Modellierung von Wachstumskurven mit ver- schiedenen nicht-linearen und linearen Wachstumsfunktionen analysiert und mit der Ge- wichtsentwicklung von Mastschweinen verglichen. Dazu wurden Gewichtsdaten der zwei dänischen Populationen benutzt. Insgesamt wurden 189’725 Gewichte von 33’704 Tieren, die in einem Altersabschnitt von der Geburt (Tag 0) bis zum 700. Lebenstag erfasst wur- den, für die Untersuchung herangezogen.
Die verschiedenen Wachstumsmodelle wurden mit dem Informationskriterium von Akaike (AIC) miteinander verglichen, wobei niedrige Werte eine gute Anpassung bedeuten. Die gesamte Wachstumskurve der beiden Populationen betrachtend, ergaben sich die niedrigsten AIC Werte für die Polynome 3. und 4. Ordnung. Unter den nicht-linearen ergab die Logistische Funktion den höchsten AIC Wert und weist somit die schlechteste Anpas- sung für die verwendeten Gewichtsdaten auf.
Ein Vergleich mit Mastschweinen zeigte, dass Minischweine eine annähernd lineare Ge- wichtsentwicklung im Alter von 0 bis 160 Tagen aufweisen. Mastschweine hingegen zei- gen sehr geringe tägliche Zunahmen im ersten Drittel der untersuchten Zeitperiode in Relation zu einem spezifischen Endgewicht. Später haben Mastschweine erhöhte tägliche Zunahmen was in einer mehr sigmoiden Wachstumskurve resultiert.
Nach der Analyse der Gewichtsentwicklung des Göttinger Minischweins wurden geneti- sche Parameter für das Gewicht geschätzt, um das Potential für eine Zucht auf niedriges Körpergewicht zu untersuchen. In einer ersten Studie wurden genetische Parameter für den Altersabschnitt von 30 bis 400 Tagen mit einem Random Regression Modell (RRM) ge- schätzt. Das RRM beinhaltete den zufälligen Effekt des Tieres, der permanenten Umwelt und der Wurfumwelt. Regressionen der zufälligen Effekte wurden mit dem Legendre
Polynom 2. Ordnung modelliert. Acht verschiedene Altersklassen wurden gebildet, um heterogene Restvarianzen berücksichtigen zu können.
Die Heritabilitäten waren moderat und lagen in einem Bereich von 0,21 (375 Tage) bis 0,25 (275 Tage). Genetische und phänotypische Korrelationen zwischen den Gewichten in unterschiedlichen Altersklassen nahmen mit zunehmender zeitlicher Distanz zwischen den Wiegungen ab, ein Umstand, auf den bei der Bildung eines neuen Zuchtschemas besondere Rücksicht genommen werden muss. Die Analyse der Eigenfunktionen zeigte, dass Selek- tion auf niedriges Körpergewicht über den gesamten untersuchten Zeitabschnitt positive Effekte auf dieses Merkmal hat.
Um detaillierte Aussagen treffen zu können und mit einem Fokus auf die Praktikabilität, wurden genetische Parameter für einen Altersabschnitt von 30 bis 700 Tagen geschätzt, um auch einen Überblick im höheren Alter zu haben. Zusätzlich wurden die Parameter mit einem Mehrmerkmalsmodell und einem Random Regression Modell geschätzt, um das geeignete Modell für eine routinemäßig ablaufende Zuchtwertschätzung zu ermitteln. Ein weiteres Ziel dieser Studie war das Auffinden der besten Selektionsstrategie für die Zucht auf ein genetisch kleineres Minischwein. Dafür wurden 19’505 Gewichte von 3’461 Mini- schweinen analysiert.
Die Heritabilitäten waren moderat, wobei die Werte des RRM etwas höher waren. Geneti- sche Korrelationen zwischen Gewichten an unterschiedlichen Tagen nahmen mit zuneh- mender zeitlicher Distanz zwischen den Messungen ab, was bereits für den Altersabschnitt 30 bis 400 Tage gezeigt wurde.
Ein Zuchtziel für die Relative Gewichtsreduzierung (RGR) wurde geschaffen, in welchem die Gewichtsreduzierung innerhalb Altersklasse in Prozent des aktuellen Körpergewichts ausgedrückt wird. Die RGR ist gewichtet in Abhängigkeit von dem Anteil der verkauften Tiere von einer Altersklasse zur nächsten. Der erwartete Zuchtfortschritt wurde für zwei verschiedene Selektionszeitpunkte (80 und 150 Tage) berechnet, wobei die Selektion am 150. Tag zu einem höheren Zuchtfortschritt im Merkmal RGR geführt hat.
Weiterhin wurden in dieser Arbeit verschiedene die Selektion betreffende Aspekte, wie z.B. der Verlust an genetischer Varianz, markergestützte Selektion und Inzuchtdepression, diskutiert.
Summary
The main focus of this thesis was the examination of a possible body weight reduction in Goettingen minipigs and the detection of an optimal selection strategy for this trait. The aims were the analysis of body weight development of the Goettingen minipig and the estimation of genetic parameters for body weight. Further, different selection strategies had to be examined to build up a new breeding scheme.
As a first step, the growth pattern of the Goettingen minipig was studied. The body weight development of minipigs was investigated by modeling growth curves with different non- linear and linear functions and by comparing their body weight development with the de- velopment of normal fattening pigs. Data of two Danish sub-populations were used. In total 189,725 body weight measurements of 33,704 animals collected from birth (d 0) to 700 d of age were analyzed.
The different growth models were compared by using the Akaike’s Information Criterion (AIC), whereas a small AIC value indicated a good fit. Regarding the whole growth curve linear polynomials of third and fourth order of fit had the smallest AIC values for the two sub-populations. Among the nonlinear functions the Logistic had the highest AIC value indicating the poorest fit.
A comparison with fattening pigs showed that the minipigs have a nearly linear body weight development in the time period from birth to 160 d. Fattening pigs have very low weight gains in the first third of the examined time period in relation to a specific end weight. Later fattening pigs have increasing daily weight gains resulting in a growth curve that is more s-shaped than the growth curve of the minipig.
After analyzing the body weight development of minipigs, genetic parameters for body weight were estimated to discover the potential for breeding on low body weight. In a first study, genetic parameters were estimated for a time period of 30 to 400 days with a ran- dom regression model (RRM). The RRM included random animal, common litter envi- ronment and permanent environment effects, respectively. Regressions for the random effects in the RRM were modeled using Legendre polynomials of second of fit. Eight age classes were built to consider heterogeneous residual variances.
The heritabilities were moderate and ranged from 0.21 (375 d of age) to 0.25 (275 d of age). Genetic and phenotypic correlations between body weights in different age classes decreased with increasing distance between age classes, a circumstance which has to be
focused on when building up a new breeding scheme. The analysis of eigenfunctions showed that a selection for low body weight has positive effects on this trait throughout the whole range of time.
To get more into detail and with a focus on practicability genetic parameters were esti- mated for a time period of 30 to 700 days to have an overview also over later ages. Further, the parameters were estimated using a multiple trait model (MTM) and a random regression model (RRM) to detect which model is more appropriate for routinely working breeding value estimation. Another aim was to find out the best selection strategy to get a genetically smaller minipig in the future. Therefore, 19,505 body weight measurements of 3,461 Goettingen minipigs were analyzed.
Heritabilities were moderate with slightly higher values estimated with the RRM. Genetic correlations between body weight measurements at different ages were decreasing with increasing time lag between the measurements as it was already shown for the time period of 30 to 400 days.
An operational breeding goal for relative weight reduction (RWR) was suggested in which the weight reduction in each age class is expressed as per cent of the actual body weight and is weighted according to the proportion of animals sold in this age class. Expected genetic progress was calculated for two different selection ages (80 and 150 d), whereas the selection at 150 d of age resulted in a higher genetic progress for RWR.
Further, several aspects related to selection like decrease of genetic variance, marker- assisted selection or inbreeding depression were discussed in this thesis.
1st CHAPTER
General introduction
General introduction
Since centuries the pig is not only known as a meat producing animal but also as an animal model for research in human medicine. Due to the influence of the Catholic Church in the High Middle Ages it was not allowed to dissect human bodies for medical research. But already that time people knew about the high similarities between humans and pigs and used pigs to investigate human diseases. Also the first book about anatomy of animals, which was written 1120 AD, is about the anatomy of pigs (Driesch and Peters, 2003).
Nowadays, pigs are widely used as laboratory animals for biomedical research. The number of used laboratory pigs increased in Germany from 1995 to 2005 from 9,500 to 14,000. However, compared to mice, which had the highest percentage in 2005 with 59 % of all used laboratory animals in Germany, pigs still have a low percentage of only 0.6 % (BMELV, 2006). Svendsen (2006) gives a number of about 60,000 pigs which are used per year for scientific research in the EU. Among pigs, miniature breeds were developed for the special demands which occur in experiments. Their body weight is much lower than that of normal pig breeds and as a consequence the costs for experiments are reduced when the test compounds are dosed per kg body weight of the recipient. A further reduction of body weight in the future could therefore be of considerable economical advantage for the research facilities.
The development of a new approved drug costs about 800 Mio. US $, that is 250 % more than 10 years ago (DiMasi et al., 2003). Preclinical tests, in which animal experiments are included, amount ca. 40 % of the total costs, whereas the proportion between preclinical and clinical tests changed so that the costs for preclinical tests are now relatively lower than for clinical studies. Even though researchers are anxious to reduce animal experiments as much as possible, these experiments are still essential to assess the effect of new drugs or techniques in the context of the whole metabolism which cannot be simulated or assessed through in vitro techniques like cell cultures. All animal experiments are underlying strict laws and official controls. In drug development, animal models are mainly used for the assessment of effectivity, pharmacokinetic studies, dosing and toxicology.
The minipig became more and more popular as a laboratory animal in the last years because of its high anatomical and physiological similarities to humans (Brandt et al., 1997). The main similarity between humans and minipigs which is attracting researchers is
the skin. The minipig epidermis has more or less the same thickness as the human epidermis. Also the cardiovascular, the intestinal and the immune system are more similar between pigs and humans than between humans and other non-rodent experimental animals like dogs or primates (McAnulty, 1999).
For toxicological studies one rodent and one non-rodent animal model has to be used. Mice and rats are the mostly used rodents, whereas rabbits are widely used as non-rodent animal model. An overview over all used laboratory animals in Germany is given in Table 1 for the year 2005. In this table, minipigs are included in the 14,004 pigs. Of all minipigs, around 1,000 Goettingen minipigs were sold to Germany in 2005 (Ellegaard Goettingen minipigs, ApS, Denmark, personal communication).
Prices of minipigs are in the same range than prices of dogs but they are much cheaper than primates. Ethical problems are much less with pigs than with primates or dogs due to the fact, that they are not caught for the special purpose of testing medical drugs or techniques as it is done with wild-living primates and that they are not kept as companions of humans like dogs (Gad, 2007).
The most important minipig breeds which are used in medical research are among others the Goettingen minipig, the Minnesota minipig, the Yucatan minipig and the Hanford minipig. Even though the Minnesota minipig was developed quite early in 1949 (Dettmers et al., 1965), the Goettingen minipig, developed in 1960’s at the University of Goettingen, Germany, is today the main minipig breed used in medical experiments. It combines the low weight and slim body shape of the Minnesota minipig, the high fertility of Vietnamese potbellied pigs and the white skin of German Landrace pigs (Glodek and Oldigs, 1981).
Besides the base population in Germany, Goettingen minipigs are bred in two populations in Denmark since 1992 and one population in the USA since 2002 in full-barrier breeding facilities to provide laboratory animals with the highest hygienic standard.
The breeding system for Goettingen minipigs differs considerably from breeding systems for meat producing pigs. Breeders of fattening pigs aim an optimization of the discounted net profit, which means to have maximum profit in the important traits like average daily gain, leanness and fast growth produced with minimum costs (Olsen and Sehested, 2000).
Table 1. Number of the different laboratory animals used in Germany in 2005, the percentage of each species in each group and over all groups
Group Species n % in each group % over all groups Mice (Mus musculus) 1,432,492 69.5
Rats (Rattus norvergicus dom.) 571,257 27.7 Guinea pigs (Cavia aperea) 40,297 2.0 Hamster (Mesocricetus auratus) 8,581 0.4 Other rodents 7,919 0.4
Rodents
Total 2,060,546 100.0 85.4
Rabbits (Oryctolagus cuniculus) 105,293 4.4
Cats (Felis catus) 1,023 3.7
Dogs (Canis lupus familiaris) 4,982 18.0 Ferrets (Mustela putorius furo) 560 2.0 Other Canines 235 0.8
Horses, donkeys etc. 755 2.7
Pigs (Sus scrofa) 14,004 50.5
Goats (Capra hircus) 283 1.0
Sheep (Ovis gmelini aries) 3,652 13.2
Primates 2,105 7.6
Other mammals 123 0.5
Non-rodent mammals (without rabbits)
Total 27,722 100.0 1.2
Quails (Coturnix coturnix) 4,159 0.2
Birds 93,858 3.9
Reptiles 153 0.01
Amphibians 16,577 0.7
Other vertebrates
Fishes 101,551 4.2
Total 2,412,678 100.0
In contrast, Goettingen minipigs, which are under breeding control of the University of Goettingen, are bred considering the demands of the customers, i.e. the researchers conducting biomedical studies. Besides a preferably high uniformity in the pigs, e.g.
concerning body weight at a certain age, skin and eye color, they should be as small as possible, show a calm temperament, reduced hair coat, useful ear veins and no abnormalities. Another specialty in Goettingen minipigs compared to normal pig breeds is
that they are not sold at a certain weight or age but they are available for the customers at all ages. This is due to the fact that all studies differ in their aims and minipigs of different ages are needed for different purposes. Most of the minipigs are used for toxicology test with an age of 3 to 6 months.
Since the development of the breed, Goettingen minipigs were phenotypically selected for low body weight on the basis of birth and weaning weight (Glodek and Oldigs, 1981).
Since the 1970’s the breeding goal focused more on the trait litter size to improve the breeding performance of sows because of a high demand for Goettingen minipigs. This resulted in a positive development of body weight due to a slightly positive correlation between the traits litter size and body weight (Ferguson et al., 1985). At the moment, breeding values are only estimated for the trait number of piglets born alive to select animals with a high breeding performance on a genetic basis. But due to an increasing need of Goettingen minipigs in medical research, it is more and more important to consider the customers’ demand of a lighter and smaller minipig. Therefore it is necessary to include the trait low body weight in the breeding goal and to select for it on the basis of breeding values.
The aims of this study were firstly the investigation of minipig growth in chapter 2 which included the detection of the best model for fitting minipig growth and the comparison of weight development of minipigs and fattening pigs to emphasize the differences in body weight development of these two breeds. Further, genetic parameters for body weight were estimated in chapter 3 using different data sets and models for analyzing the genetic background of this trait. Finally, in chapter 4 a new breeding scheme was developed to improve the weight reduction in the future considering different selection strategies.
References
BMELV. 2006. Verbraucherministerium. www.verbraucherministerium.de.
Brandt, H., B. Möllers, and P. Glodek. 1997. Prospects for a genetically very small minipig. Pages 93-96 in Satellite Symposium to Eurotox '97: The Minipig in Toxicology. ed. O. Svendsen, Aarhus, Denmark.
Dettmers, A. E., W. E. Rempel, and R. E. Comstock. 1965. Selection for small size in swine. J. Anim Sci. 24: 216-220.
DiMasi, J. A., R. W. Hansen, and H. G. Grabowski. 2003. The price of innovation: new estimates of drug development costs. Journal of Health Economics 22: 151-185.
Driesch, A. v. d., and J. Peters. 2003. Geschichte der Tiermedizin - 5000 Jahre Tierheilkunde. Schattauer, Stuttgart.
Ferguson, P. W., W. R. Harvey, and K. M. Irvin. 1985. Genetic, phenotypic and environmental relationships between sow body weight and sow productivity traits.
J. Anim. Sci. 60: 375-384.
Gad, S. C. 2007. The minipig. Pages 731-771 in Animal models in Toxicology. ed. S. Gad, Taylor and Francis Group, Boca Raton.
Glodek, P., and B. Oldigs. 1981. Das Göttinger Miniaturschwein. Paul Parey, Berlin.
McAnulty, P. A. 1999. The value of the minipig in toxicity and other studies supporting the development of new pharmaceuticals. European Pharmaceutical Contractor: 82- 86.
Olsen, D., and E. Sehested. 2000. A method of modelling pig breeding. Acta Agric. Scand.
50: 1-11.
Svendsen, O. 2006. The minipig in toxicology. Experimental and Toxicologic Pathology 57: 335-339.
2nd CHAPTER
Modeling the growth of the Goettingen minipig
F. Köhn, A. R. Sharifi, and H. Simianer
Institute of Animal Breeding and Genetics, University of Göttingen, 37075 Göttingen
Published in
Journal of Animal Science (2007) 85: 84-92
Modeling the growth of the Goettingen minipig
F. Köhn, A. R. Sharifi, and H. Simianer
Institute of Animal Breeding and Genetics, University of Göttingen, 37075 Göttingen
Abstract
The Goettingen minipig developed at the University of Goettingen, Germany, is a special breed for medical research. As a laboratory animal it has to be as small and light as possi- ble to facilitate the handling during experiments. For achieving the breeding goal ‘small body size’ in the future, the growth pattern of the minipig was studied. This study deals with the analysis of minipig body weight development by modeling growth curves with different non-linear and linear functions and the comparison to the body weight develop- ment of normal fattening pigs. Data were provided by Ellegaard Goettingen minipigs, Denmark, where two sub-populations of the Goettingen basis population are housed. In total 189,725 body weight recordings of 33,704 animals collected from birth (d 0) to 700 d of age were analyzed. Seven different non-linear growth functions and four different poly- nomial functions were applied. The different growth models were compared by using the Akaike’s Information Criterion (AIC). Regarding the whole growth curve linear polynomi- als of third and fourth order of fit had the smallest AIC values indicating the best fit for the used minipig body weight data. Among the nonlinear functions the Logistic had the highest AIC value. A comparison with fattening pigs showed that the minipigs have a nearly linear body weight development in the time period from birth to 160 d. Fattening pigs have very low weight gains in their first 7 weeks in relation to a specific end weight. After 7 weeks fattening pigs have an increasing body weight development resulting in a growth curve that is more s-shaped than the growth curve of the minipig. Based on these results further studies can be conducted to analyze the growth with random regression models and to estimate variance components for optimizing the strategies in minipig breeding.
Keywords
Body weight development, growth curve, growth model, Goettingen minipig
Introduction
The Goettingen minipig is an important laboratory animal, especially used for the safety assessment of new pharmaceuticals and for surgery purposes and disease models, because of shared anatomic and physiologic characteristics with humans (Brandt et al., 1997). The breed was developed in the 1960s at the University of Goettingen, Germany. After 30 years of successful breeding with a focus on breeding goals like small body size, unpig- mented skin and adequate fertility, 38 pregnant sows were brought to the Ellegaard farm in Denmark in 1992 to build up a new population. Today there are about 3,500 Ellegaard Goettingen minipigs under breeding control of the University of Goettingen.
The main difference between a minipig and a normal sized pig is the smaller body size of the minipig. Goettingen minipigs should have a mature body weight of 35 to 45 kg (Bollen et al., 1998). This small body size is advantageous for the reduction of rearing and breed- ing costs and for a comfortable handling of the minipigs during medical research. It can also be assumed that not only the mature body weight of minipigs differs from fattening pigs but also the characteristics of body weight development as far as the fattening pigs have been selected for fast growth in the last two-thirds of the fattening period. In compari- son to fattening pigs the growth of minipigs should be slow, mainly in the first months after birth so that the minipigs have a low body weight when entering laboratory facilities with an average age of 3 to 6 months.
Knowing the growth pattern of minipigs is necessary for later genetic research that will be conducted on this issue, e.g., estimation of variance components for growth at a certain age. The aims of this study were the application of different non-linear and polynomial functions to describe the body weight data of minipigs and the investigation of the differ- ence in body weight development to fattening pigs. Results are prerequisites for the appli- cation of random regression models for estimating genetic parameters.
Data and Methods
Growth Curves Modeled over all Data
Data Description. For the analysis of growth, the data were provided from Elle- gaard Goettingen minipigs with a total number of 199,764 body weight records of 33,749 animals acquired from 1995 to 2005. In this time period there was no organized selection
for body weight. Minipigs were selected for breeding because of low inbreeding coeffi- cients and adequate exterior traits like white skin and little hair coat. Due to scarcity, body weight data measured after 700 d of age were excluded from the analyses. Outliers of the data set were detected using the influence diagnostics as recommended by Belsley et al.
(1980). With this method the influence of each observation on the estimates is measured.
Influential observations are those that appear to have a large influence on the parameter estimates. The method of Belsley et al. (1980) is incorporated in the SAS procedure REG by using the INFLUENCE option in the MODEL statement. The studentized residual was used for analyzing the influence of each weight record. The studentized residual was calculated as follows:
(
i)
i i
h s
RSTUDENT r
= −
) 1
(
where ri =yi−yˆi, s(i)²is the error variance estimated without the ith observation and hi is the hat matrix, which is the ith diagonal of the projection matrix for the predictor space.
Belsley et al. (1980) suggest paying special attention to all observations with RSTUDENT larger than an absolute value of 2. We discarded all records lying outside the 95 %-confi- dence interval [-1.99; 2.15]. This resulted in a total number of 189,725 body weight records of 33,704 animals that were used for the growth analysis.
The minipigs were weighed routinely without a special treatment like starvation before weighing. The animals were weighed routinely at various intervals. On average there have been 5.63 body weight recordings per minipig with a maximum number of 23 body weight recordings (Figure 1). Nearly all animals were weighed on their day of birth (d 0). Pigs were then weighed at weaning (21 to 28 d of age) and then again at 8 weeks of age, when they left the rearing unit. Later all minipigs were weighed once each month and each minipig was weighed before it was sold.
On the Ellegaard farm the minipigs are housed in two different units. Minipigs from both units have the same ancestors, but there is no genetic exchange between the units. Thus, every unit is an independent sub-population with a specific environment. Because of these circumstances the calculations were made for each unit separately. It is then possible to detect potential differences in the growth curves between unit 1 and unit 2 due to environ- mental effects.
Weight recordings per animal
n animals
3315
818 2137
58376072
4586
3160 2600
1961 1252
820 497337
161 77 39 11 10 6 2 2 3 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0
1000 2000 3000 4000 5000 6000
Figure 1. Number of body weight recordings per animal and corresponding number of animals for all minipigs used in this study
Within every unit the growth curve was modeled for each sex. The detailed numbers of analyzed animals and body weight records are shown in Table 1.
Table 1. Number of animals and records per sex and unit used in the calculations
Sex Unit1 Animals Records
Male 1 10,717 59,045
2 6,676 37,012
Female 1 9,848 55,556
2 6,463 38,112
Total 33,704 189,725
1independent sub-population.
There were no differences in the feeding regime in both units. The minipigs were provided a special minipig diet with 10.0 MJ ME/kg and 13.9 % CP of 250 to 500 g/d progressively fed from 2 to > 12 months of age to males. Females were fed approximately 80 % of the amount offered to males.
Growth Models. To estimate the body weight at a certain age four 3-parameter and three 4-parameter non-linear growth functions as well as 4 different polynomial growth functions were fitted to the minipig body weight data. The equations for the applied growth models are given in Table 2.
Table 2. Functions considered in this study for modeling the growth curve of the minipig Model Equations* Parms# Reference Logistic
( )
(
bae c t)
W − ×
×
= + 1
3 (Fekedulegn et al., 1999) Gompertz W = a × e−e(b−(c×t)) 3 (Wellock et al.,
2004)
v. Bertalanffy
3 3 1
3 1 0
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟×
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−
−
= e− ×b×t
W b
a b
W a 3 (Bertalanffy,
1957b)
Brody W =a×
(
1−b×e(−c×t))
3 (Fitzhugh Jr., 1976)Richards
( )
(
b e c t)
mW a 1
1+ × − ×
= 4 (Fekedulegn et
al., 1999) Bridges W =W0 +a×
(
1−e(−m×tp))
4 (Wellock et al.,2004)
Janoschek W = a−(a−W0)×e(−c×tm) 4 (Wellock et al., 2004)
Ali-Schaeffer 2
4 3
2 2 1
0
ln700 ln700
700
700 ⎟
⎠
⎜ ⎞
⎝ + ⎛
⎟⎠
⎜ ⎞
⎝ + ⎛
⎟⎠
⎜ ⎞
⎝ + ⎛
⎟⎠
⎜ ⎞
⎝ + ⎛
= d t
d t d t
d t d
W 5 (Ali and
Schaeffer, 1987) Polynomials
∑
=× +
= r
i
i
i t
d d
W
1
0 3 to 5 (Hadeler, 1974)
*W = body weight; W0 = initial weight of the animal in kg; a = mature weight of the animal in kg; t = age in days; b, c, m, p = parameters specific for the function; r = second to fourth order of fit; d0 = intercept; di = regression coefficients.
#number of parameters.
The Logistic (Robertson, 1908) as well as the Gompertz function (Gompertz, 1825) were developed in former centuries. They have three parameters in the equation with no flexible point of inflection. The von Bertalanffy function (Bertalanffy, 1957a), developed in 1957, is also a 3-parameter equation where the point of inflection lies at 30 % of the mature body
weight. A critical assumption of this function is that the anabolic processes of the body are proportional to the surface of the organism and the catabolic processes are proportional to the body weight. These conditions are not necessarily given in the growth of animals (Schönmuth and Seeland, 1994). The 3-parameter Brody function from 1945 also has a priori no point of inflection (Brody, 1945). The 4-parameter Richards function was devel- oped in 1959 as an advancement of the Logistic and the Gompertz function (Richards, 1959). It has a flexible point of inflection and thus is suitable for the application to animal growth. The Janoschek function from 1957 (Janoschek, 1957) and the Bridges function from 1986 (Bridges et al., 1986) are also very flexible in their points of inflection and are mainly used to describe the postnatal growth of an individual. The Ali-Schaeffer function (Ali and Schaeffer, 1987) can be interpreted as a fractional polynomial of degree 4 which is mostly used in test day models for calculating lactation curves of dairy cows. Guo (1998) classified the Ali-Schaeffer function as a function of the mixed-log family. Linear models with a polynomial structure from second up to fourth order of fit were applied.
Higher orders of fit did not achieve a significant influence on the fit of the growth curve (P < 0.001, F-Statistic, SS Type 1) and were therefore not considered in the analysis.
Statistics and Model Comparison. The estimation of the non-linear growth curves was carried out using the NLMIXED procedure of the SAS System Version 9 (SAS, 2002). The linear growth curves were calculated with the MIXED procedure of the same software. The models were compared by using Akaike’s Information Criterion (AIC) (Akaike, 1973):
AIC = -2 logLikelihood + 2(number of parameters).
This information criterion is useful for comparing models with different numbers of parameters. It is therefore more advantageous than the R² which increases with increasing numbers of parameters in the models and thus, is not useful for the comparison of models with different numbers of parameters. The comparison of different non-nested models with this method determines which model is more likely to be correct and accounts for differ- ences in the number of degrees of freedom. By using the ML method in the MIXED procedure, the AIC values obtained from both the MIXED and the NLMIXED procedure can be compared. The model with the smallest AIC value was chosen to be the best for fitting minipig body weight data.
For the model reduction from fourth to second order of fit in the linear polynomial functions the F-statistic was used.
Comparison Minipig – Fattening Pig
The results from Kusec (2001) were used for comparing the growth of the minipig with the growth of a fattening pig. This author analyzed the growth of 72 4-line crossbred barrows from an age of 9 to 26 weeks. The pigs weighed 23 kg at the beginning of the test and 138 kg (intensively-fed pigs) and 117 kg (restrictively-fed pigs) at the end of the experiment.
The barrows had a Piétrain x Hampshire sire and a Large White x German Landrace dam.
This crossbreed represents standard fattening pigs of the German Hybrid Pig Breeding Programme (Bundes-Hybrid-Zuchtprogramm, BHZP). The pigs in the experiment were fed in two different groups, an intensively-fed group and a restrictively-fed group. Addition- ally, two different genotypes of malignant hyperthermia-susceptible (MHS) pigs were examined within a feeding group. The MHS-gene has a positive effect on the carcass lean- ness but a negative effect on meat quality. The pig breeders have the option to choose sire lines of different MHS-gene status, whereas the dam lines in Germany are completely MHS-negative. The genotypes of the fattening pigs used in the study of Kusec (2001) were MHS-carrier (Nn) and MHS-negative pigs (NN). Both are often used genotypes in cross- bred fattening pigs in Germany. In the study of Kusec (2001) the growth curve for body weight was modeled with the Richards function so it can be compared with the growth curve of minipigs estimated with the Richards function in this study. On the basis of the estimated parameter values (Table 3) the body weight for fattening pigs were calculated for 0 to 700 d of age. For this calculation the parameter estimates of the Nn-genotype were used because there was no significant (P > 0.05) difference between the parameter values of the two MHS-genotypes (G. Kusec, Faculty of Agriculture, Osijek, Croatia, personal communication).
First, the body weight of minipigs and fattening pigs at d 160 (160d-W) were compared.
The 160th d is a typical day for slaughtering fattening pigs in performance tests in Germany.
Table 3. Parameter values of the Richards function for fattening pigs (Kusec, 2001) and minipigs (males of unit 1) used in this study
Fattening pigs
Parameter1 Intensively fed Restrictively fed Minipigs
a 220 160 53.47
b 0.054476 0.057374 -0.9628
c 1.38986 1.688707 0.00202
m 0.01 0.01 -0.7507
1a = mature body weight, kg; b = biological constant; c = maturing index; m = shape parameter determining the position of the inflection of the curve point.
The 160d-W from minipigs and fattening pigs was set to 100 % to achieve comparability.
Second, the age was detected at which the pigs weigh 40 % of the predicted mature body weight. Again the particular body weight at 40 % of maturity (40%-MW) was set to 100 % for the minipigs and the fattening pigs to enable a comparison between the breeds.
To simplify the presentation only the results of male minipigs from unit 1 are used for the comparison of the minipigs with fattening pigs. Results for males in unit 2 and females in both units are not shown, but they were very similar to the results of males in unit 1.
Results
Growth Curves Modeled over all Data
Comparing the models by AIC values and the residuals showed the following results (Table 4). The polynomial function of third order of fit had the smallest AIC values for both sexes in unit 1. For unit 2 the smallest AIC values were achieved of the polynomial of fourth order of fit.
The examination of the polynomial models from second to fourth order of fit using the F- Statistic SS type 1 showed that the polynomials of second and third order of fit were sig- nificant (P < 0.01) in unit 1 whereas the polynomial of fourth order of fit had no significant influence on the estimation of the growth curves for both sexes in unit 1. In unit 2 the polynomial of fourth order of fit had a significant influence (P < 0.01) on the fit of the growth curve. Thus, the growth curves for unit 1 for both sexes were best fitted using a polynomial of third order of fit and the growth curves for unit 2 were best fitted by a poly- nomial of fourth order of fit.
Table 4.Values of -2 Lm (Lm is the maximized log-lokelihood), Akaike’s Information Criterion (AIC), and residual variances (res.) for male and female minipigs of both units1 Sex MaleFemale Unit Unit12 Unit22 Unit12 Unit22 Criterion/ Model -2 LmAIC Res. AIC Res. AIC Res. AIC Res. -2Lm-2 Lm-2Lm Logistic 300,146 300,152 2 7 4 3.245 188,756 188,763.262 299,771 299,773.558 209,908 209,913.670 Gompertz 259,874 259,880 3 2 1 2 2 3 2 1 8 2 3 4 3 5 7 3 4 9 8 4 6 2 5 8 0 9 1 4 3 1 1 6 8 0 2
2.563 162,297 162,302.547 257,226 257,232.792 176,895 176,902.804 Bertalanffy 248,756 248,762.375 154,826 154,832.345 245,587 245,592.583 168,356 168,362.580 Brody 245,935 245,942.327 152,812 152,812.291 242,906 242,912.534 169,792 169,802.617 Richards 242,096 242,102.262 150,195 150,202.220 238,687 238,692.458 164,319 164,322.474 Bridges 242,415 242,422.268 150,346 150,352.224 238,931 238,932.463 164,480 164,482.478 Janoschek 246,106 246,112.330 152,588 152,592.285 243,274 243,282.541 168,597 168,602.586 Ali-Schaeffer 240,068 240,072.228 149,260 149,272.195 236,779 236,782.424 163,251 163,262.446 Polynomial, second order217,176 217,182.319 135,475 135,482.284 208,923 208,932.612 144,233 144,242.622 Polynomial, third order214,826 214,836 2.227 134,456 134,462.214 206,651 206,661 2.415 142,616 142,626 2.470 Polynomial, fourth order 214,826 214,832.246 134,27134,282 2.204 206,650 206,662.453 142,368 142,380 2.455 1 The lowest values for AIC and res. are printed in boldface in each column. 2 Unit 1 and unit 2 represent independent subpopulations.
The smallest AIC value for the non-linear models was calculated for the 4-parameter Richards function in both units. If a minipig growth curve is modeled with a 3-parameter model, the Brody function will be implemented according to the AIC value. The Logistic function was definitely the model with the greatest AIC value. A graphical comparison of the Logistic, the Brody and the Richards function as well as the polynomial of third order of fit is given in Figure 2. In this figure only the results for males in unit 1 are displayed as an example without loss of generality, since curves of females and for males in unit 2 look very similar.
Age, d
BW, kg
0 100 200 300 400 500 600 0
5 10 15 20 25 30 35 40 45 A
Age, d
BW, kg
0 100 200 300 400 500 600 0
5 10 15 20 25 30 35 40 45 B
Age, d
BW, kg
0 100 200 300 400 500 600 0
5 10 15 20 25 30 35 40 45 C
Age, d
BW, kg
0 100 200 300 400 500 600 0
5 10 15 20 25 30 35 40 45 D
Figure 2. Growth curves of the Goettingen minipig for males in unit 1 as predicted by the Logistic (A), Brody (B), Richards (C), and third order polynomial functions (D)
By comparing all models on the basis of residuals the best fitting model for unit 2 was not the polynomial of fourth order of fit, but the Ali-Schaffer function. For unit 1 the polyno- mial of third order of fit had the lowest residuals. The Brody function was the best fitting 3-parameter function and the Richards function was the best fitting 4-parameter function on the basis of the calculated residuals for both units.
Comparison Minipig – Fattening Pig
According to the calculated body weight for the fattening pigs and the predicted body weight for the minipigs from the Richards function, Figure 3 shows the relative body weight development of the pigs from birth to 160 d of age. The minipigs have an almost linear body weight development in this time period. The fattening pigs show very small body weights in relation to their particular 160d-W up to 50 d of age. Then after 50 d of age they have increasing body weights through to 160 d of age. The restrictively-fed pigs are characterized by a steeper slope of the curve after d 42 in relation to the intensively-fed pigs. Thus, compared to the intensively-fed pigs, they have greater ADG after 42 d of age.
Age, d
Percentage of 160-d BW
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 0
10 20 30 40 50 60 70 80 90 100
Fattening pig intensive Fattening pig restricive Minipig
Figure 3. Relative body weight development of male minipigs of unit 1 and normal, fattening pigs from birth weight to 160-d body weight (160d-W)
The birth weights of the analyzed breeds differ in a remarkable way in relation to the 160d-W. At birth and throughout birth to 160 d of age, the proportion of body weight to its 160d-W is greater in the minipig compared with the fattening pig.
The second comparative analysis determined the body weight development of minipigs and fattening pigs from birth to a stage of 40 % of maturity. The body weights and ages for minipigs and fattening pigs are shown in Table 5.
Table 5. Mature body weight and ages at 40% of mature body weight (40%-MW) of normal, fattening pigs and minipigs (males of unit 1) calculated with parameters of the Richards function
Fattening pigs1
Item Intensively fed Restrictively fed Minipigs Mature body weight, kg 2201 1601 53
40%-MW, kg 88 64 21
Age at 40%-MW, d 129 109 310
1Kusec (2001)
The minipigs need 2.4 times longer than the intensively-fed fattening pig to reach 40%- MW. In relation to their 40%-MW the minipigs have a greater birth weight than the fattening pigs. It can be observed that the minipigs have a nearly linear body weight devel- opment until the chosen degree of maturity (data not shown). Further, the minipigs can realize greater ADG from birth up to 50 % of age at 40% maturity in relation to the 40%- MW than fattening pigs. After 50 % of age at 40% maturity the body weight development of the fattening pigs increases more rapidly. Restrictively-fed fattening pigs have in all stages until 40 % of maturity a lower body weight in relation to their 40%-MW than the intensively-fed fattening pigs.
Discussion
Modeling growth curves of animals is a necessary tool for optimizing the management and the efficiency of animal production. It is obvious that growth modeling has many advan- tages for meat producing animals (Schinckel and de Lange, 1996). As a consequence, many studies dealing with modeling of growth curves have used pigs (Bastianelli and Sauvant, 1997; Knap et al., 2003; Krieter and Kalm, 1989; Schinckel et al., 2003; Wellock et al., 2004; Whittemore, 1986), cattle (Brown et al., 1976; Lopez de Torre et al., 1992;
Menchaca et al., 1996), and poultry (Knizetova et al., 1991b; Sengül and Kiraz, 2005). In the last few years it has become more and more popular to analyze the growth of special livestock, e.g., the Bolivian llama (Wurzinger et al., 2005) and the pearl gray guinea fowl (Nahashon et al., 2006) for providing improvement for their husbandry. Because of the increasing importance of the Goettingen minipig for medical research it is necessary to
analyze the body weight development of this breed for the well-founded derivation of required space for guidelines of laboratory animals and adjusted feeding regimens. The knowledge of growth pattern in minipigs also facilitates the creation of experimental designs.
The growth of the Goettingen minipig is not the same as the growth of pig breeds for meat production. Apart from the different mature body weight the body weight development of the minipig should be slower than the body weight development of normal pigs, mainly in the first year so that the mature body weight of the Goettingen minipig will be achieved later than the mature body weight of normal pigs. On the basis of growth parameters of the Richards function estimated in this study and in the study of Kusec (2001) the age at maturity was calculated for minipigs and intensively-fed fattening pigs. Minipigs are expected to have a constant mature body weight around 5 years of age while intensively- fed fattening pigs reach a constant body weight with approximately 4.2 years of age. Thus, the minipigs seem to reach maturity at an older age than fattening pigs. Body measure- ments, such as length and circumference of skeletal tissue, provide a more exact determi- nation of maturity (Lawrence and Fowler, 2002) and these data should be included when there is a focus on size at maturity.
The application of different non-linear and linear functions to body weight data from Goettingen minipigs is the first study made for this species in such detail. The data set of the current study is unique based on the very large number of animals and body weight recordings over a wide range of ages. The results of the non-linear functions showed that the Richards function was the best for fitting the minipig data according to the AIC values.
This is in agreement with Brown et al. (1976) who applied the Logistic, Gompertz, von Bertalanffy, Brody, and Richards function to body weight data of different cattle breeds.
Brown et al. (1976) also noted that the Logistic model is worst for fitting their cattle body weight data. The Richards function was also considered as best by Knizetova et al. (1991a) for growth analyses in ducks and other poultry, like geese and chicken, after checking the Logistic, Gompertz and Richards function for the growth of poultry in other studies.
Krieter and Kalm (1989) applied the growth model developed from Schnute (1981) for estimating the growth curves of Large White and Piétrain pigs. The pigs were weighed weekly from 25 kg to 223 kg (Large White) and from 29 kg to 186 kg (Piétrain), respec- tively. In the model proposed from Schnute (1981) the non-linear growth functions Logistic, Gompertz, von Bertalanffy, Richards, and others like a linear, quadratic or expo-
nential function are included. Combining several models into a single model reduces the problem of comparing the different parameters used by each individual model. Intercepts, inflection points, and asymptotic size are no longer essential, although they can be identi- fied easily if they occur. Krieter and Kalm (1989) found that the von Bertalanffy model had the lowest residual standard deviation compared with the Richards and the Gompertz function, which is not in agreement with the results of our study. A focus on the residuals of the non-linear models suggests that the Richards function is the best fitting model for minipig body weight data. Regarding all models, the polynomial of third order of fit was the best fitting model for the minipig body weight data from unit 1 and the Ali-Schaeffer function was the best model for unit 2.
The difficulty in comparing the results of this study with other studies dealing with pig growth lies in the different time periods that were examined. For minipigs the whole period of growth from birth to later ages is of interest, because the body weight at later ages is an important variable for breeding selection. Meat producing breeds like the Large White or Piétrain are observed from 30 to 105 kg body weight in the performance test, i.e., from 90 d of age at the beginning of the fattening period to 160 to 180 d of age at the end of the fattening period (ZDS, 2004). Castrated Large White pigs average 160 d of age at the end of the fattening period (VIT, 2005). Thus, the time between weaning and slaughter at 160 d is the best time to measure growth rate in fattening pigs.
An extreme example is the study of Schinckel et al. (2003), who modeled the growth of pigs from birth to 60 d of age. The authors also applied the Gompertz and the Bridges functions, additionally the Michaelis-Menten equation and an exponential linear quadratic equation of form Wij = exp (b0 + b1t + b2t²) + eij, where Wij is body weight, t is age, b0, b1
and b2 are regression coefficients and eij is the residual term. The different models were compared by using the residual standard deviation. The authors concluded from their results that the exponential linear quadratic equation is the best fitting model for estimating the growth of pigs from birth to 60 d of age. However, the authors mentioned a lack of fit for the exponential equation, resulting in an inaccurate prediction of the birth weight of each pig, and that probably a more complex function is needed.
When modeling growth of the fattening pig numerous studies deal with the pattern of com- positional growth (Bastianelli and Sauvant, 1997; Whittemore, 1986). In this case the growth curves for protein growth or lean growth, respectively, and fat growth are modeled separately (Knap et al., 2003), often combined with different levels of feed intake. There-
fore the growth curve of body weight in the time period of interest has to be modeled. The relationship of the body component (e.g., protein) to the body weight has to be fitted.
Development of growth curves for body components (e.g., protein and fat) can be a useful tool for management of pork production, because it can aid in determining the best time to slaughter animals and depending on genotype, feed intake can be predicted (Schinckel and de Lange, 1996).
Modeling different growth fractions separately is not necessary for the minipig. The minipig is a laboratory animal and is never used for commercial meat production. If body components were modeled separately it could be used to adjust the feeding regimen (e.g., feeding restrictions) to prevent the minipig from becoming too heavy.
Using estimates from the Richards function, a difference in body weight development between minipigs and fattening pigs is observed. From d 0 to 160 d of age, the minipig growth curve is approximately linear (Figure 4). This result is consistent with the minipig growth analysis conducted by Brandt et al. (1997) who modeled the growth of 283 minipig sows from birth to 1,100 d of age with the Boltzmann function. In contrast, the fattening pigs have very low ADG in the beginning and greater ADG from d 50 to the end of the time period. Thus, the growth curve of a fattening pig is s-shaped. Applications of other growth functions apart from the Richards function lead to similar results as far as these functions have a flexible point of inflection. All functions show a linear growth of minipigs in the first year except the Logistic function which has a fixed point of inflection. Thus, it is also expected that the trajectory of the growth curve of fattening pigs is s-shaped when modeling the growth with growth functions other than the Richards function as far as a flexible point of inflection exists.
It is obvious that the selection on greater ADG in fattening pigs focuses on the last 67 % of the period from d 0 to 160 d of age. The minipig growth curve should be changed into a lower level (i.e., the asymptotic body weight should be lower). A change from linearity to a more s-shaped trajectory would be advantageous for the first part of the growth curve. If this aim could be achieved, it would be possible to produce minipigs with very small ADG and as a consequence smaller body weight in the first few months of their lives, which is the main period for starting medical or pharmaceutical experiments. On the basis of genetic analyses that will be conducted the best time for selecting minipigs for lower body weight will be determined. Additionally, it could be determined if a change of the minipig growth curve is possible.
The modeling of growth curves of the Goettingen minipig with different non-linear and linear functions is a useful tool for the derivation of space and food requirements at differ- ent ages. Based on the current study, it is also possible to apply the best fitting models (i.e., linear polynomials) for the estimation and analysis of genetic parameters with random regression models concerning the breeding goal ‘low body weight’, which should be obtained especially in the juvenile period which is the main period for marketing.
References
Akaike, H. 1973. Information theory and an extension of the maximum likelihood princi- ple. Pages 267-281 in 2nd International Symposium on Information Theory, Buda- pest, Hungary.
Ali, T. E., and L. R. Schaeffer. 1987. Accounting for covariances among test day milk yields in dairy cows. Can. J. Anim. Sci. 67: 637-644.
Bastianelli, D., and D. Sauvant. 1997. Modelling the mechanisms of pig growth. Livest.
Prod. Sci. 51: 97-107.
Belsley, D. A., E. Kuh, and R. E. Welsch. 1980. Regression diagnostics: identifying influ- ential data and sources of collinearity. Wiley, New York.
Bertalanffy, L. v. 1957a. Quantitative laws for metabolism and growth. Quarterly Reviews of Biology 32: 217-232.
Bertalanffy, L. v. 1957b. Wachstum. Pages 1-68 in Handbuch der Zoologie. ed. J. G.
Helmcke, H. v. Lengerken and G. Starck, W. de Gruyter, Berlin.
Bollen, P., A. Andersen, and L. Ellegaard. 1998. The behaviour and housing requirements of minipigs. Scand. J. Lab. Anim. Sci. Suppl. 25: 23-26.
Brandt, H., B. Möllers, and P. Glodek. 1997. Prospects for a genetically very small minipig. Pages 93-96 in Satellite Symposium to Eurotox '97: The Minipig in Toxi- cology. ed. O. Svendsen, Aarhus, Denmark.
Bridges, T. C., L. W. Turner, E. M. Smith, T. S. Stahly, and O. J. Loewer. 1986. A mathe- matical procedure for estimating animal growth and body composition.
Transactions of the American Association of Agricultural Engineers 29: 1342- 1347.
Brody, S. 1945. Bioenergetics and growth. Reinhold Publishing, New York.
Brown, J. E., H. A. Fitzhugh Jr., and T. C. Cartwright. 1976. A comparison of nonlinear models for describing weight-age relationships in cattle. J. Anim. Sci. 42: 810-818.
